In general topology and related areas of mathematics, the final topology (or coinduced, weak, colimit, or inductive topology) on a set ${\displaystyle X,}$ with respect to a family of functions from topological spaces into ${\displaystyle X,}$ is the finest topology on ${\displaystyle X}$ that makes all those functions continuous.

The quotient topology on a quotient space is a final topology, with respect to a single surjective function, namely the quotient map. The disjoint union topology is the final topology with respect to the inclusion maps. The final topology is also the topology that every direct limit in the category of topological spaces is endowed with, and it is in the context of direct limits that the final topology often appears. A topology is coherent with some collection of subspaces if and only if it is the final topology induced by the natural inclusions.

The dual notion is the initial topology, which for a given family of functions from a set ${\displaystyle X}$ into topological spaces is the coarsest topology on ${\displaystyle X}$ that makes those functions continuous.

Given a set ${\displaystyle X}$ and an ${\displaystyle I}$-indexed family of topological spaces ${\displaystyle \left(Y_{i},\upsilon _{i}\right)}$ with associated functions $${\displaystyle f_{i}:Y_{i}\to X,}$$ the final topology on ${\displaystyle X}$ induced by the family of functions ${\displaystyle {\mathcal {F}}:=\left\{f_{i}:i\in I\right\}}$ is the finest topology ${\displaystyle \tau _{\mathcal {F}}}$ on ${\displaystyle X}$ such that $${\displaystyle f_{i}:\left(Y_{i},\upsilon _{i}\right)\to \left(X,\tau _{\mathcal {F}}\right)}$$

is continuous for each ${\displaystyle i\in I}$. The final topology always exists, and is unique.

Explicitly, the final topology may be described as follows:

The closed subsets have an analogous characterization:

The family ${\displaystyle {\mathcal {F}}}$ of functions that induces the final topology on ${\displaystyle X}$ is usually a set of functions. But the same construction can be performed if ${\displaystyle {\mathcal {F}}}$ is a proper class of functions, and the result is still well-defined in Zermelo–Fraenkel set theory. In that case there is always a subfamily ${\displaystyle {\mathcal {G}}}$ of ${\displaystyle {\mathcal {F}}}$ with ${\displaystyle {\mathcal {G}}}$ a set, such that the final topologies on ${\displaystyle X}$ induced by ${\displaystyle {\mathcal {F}}}$ and by ${\displaystyle {\mathcal {G}}}$ coincide. For more on this, see for example the discussion here. As an example, a commonly used variant of the notion of compactly generated space is defined as the final topology with respect to a proper class of functions.